New dispersion relations and their application to partial-wave amplitudes

Abstract

After reviewing the commonly used dispersion relations, a systematic investigation of more generalized dispersion relations on parametrized curves in the Mandelstam plane fors-u crossing-symmetric amplitudes is made with the aim of obtaining dispersion relations which receive contributions from all three channels, however, in such a way that knowledge of the absorptive parts is only required in regions well inside the various Lehmann ellipses. In addition we require that the dispersion relations receive no contributions from kinematic singularities arising from the parametrization and that they allow partial-wave projections to be made in a relatively simple manner. It is found that dispersion relations on hyperbolic curves in the Mandelstam plane are a natural solution of the problem. The dispersion relations are written in a remarkably simple form similar to the usual fixed-t dispersion relation but with an additionalt-channel contribution. As an interesting application, we derive generalized partial-wave dispersion relations for elastic pion-nucleon scattering, where the left-hand cut contribution is explicitly given by convergent partial-wave series in the crossed channels.